Statistics and the Kennedy Assassination witnesses

I am having an interesting discussion on a newsgroup (alt.assassination.jfk) regarding the single bullet theory (SBT) from the JFK assassination. The problem with the SBT is that it conflicts with much of the evidence.

The current SBT to which most SBT adherents subscribe (e.g Gerald Posner in his book Case Closed) has the first shot missing the limo and its occupants entirely at time t=160 and the second shot hitting both JFK and Gov. Connally at time t= 224 (each unit is one frame of the Zapruder film which were taken at 18.3 frames/second). During this period from t=160 to t=196 JFK is smiling and waving to the crowd. From t=197-207 he still has his hand up but is looking forward and the facial features are blurred. From t=207 to t=224 he is behind a road sign and not visible to Zapruder's camera.

The problem is that there are at least 17 witnesses by my count (their evidence is compiled http://www.dufourlaw.com/jfk/first_shot_hit_witnesses.PDF" ) who said that JFK reacted to the first shot. Most said he moved to the left and brought his hands to his neck but NONE said he kept smiling and waving to the crowd.

Also, the shot pattern according to this SBT is 1.....2.......3, the space between the first two being about 64 frames and between the last two 99 frames. However, the witnesses heard a quite different shot pattern with the last two shots closer together. I have compiled their evidence http://www.dufourlaw.com/jfk/shot_pattern_evidence.pdf".

Now, without knowing anything about the reliability of these witnesses I say that the convergence cannot reasonably happen by chance.

With respect to the 17 witnesses to 0 who said that JFK reacted to the first shot, let's say that the odds were 1/2 that any witness would get this observation wrong (quite apart from the odd fact that they all thought he reacted in the same or very similar way). The chance that in 17 witnesses you would not find one witness who said that JFK kept smiling and waving after the first shot is [itex]p = 1/2^17 = 1/131072[/itex]. And that, as I say, assumes that the witnesses were only 50% reliable.

With respect to the 44 witnesses who recalled the 1.........2...3 pattern versus the 6 who thought the pattern was 1.....2.........3, which could fit the SBT, lets say the probability of getting this right was 50%. Then the probability of 44 of the 50 witnesses hearing one pattern over the other is given by a simple binomial distribution function:

[tex]f_p(44) = \frac{50!.5^{50}}{44!6!}[/tex]

which works out to 1.4 x 10-8. For these conditions, the mean number of events is 25 and the standard deviation is 3.5.

This tells me that the witnesses are reliable because our assumption that they were not reliable gives a result that is extremely improbable. Am I wrong?

This tells me that the witnesses are reliable because our assumption that they were not reliable gives a result that is extremely improbable. Am I wrong?

Not quite. If you know what actually happened, then you can make a statement about the reliability of the witnesses. Otherwise, their reliability is just a parameter:

First we need to define reliability more precisely. Let's define reliability as the probability that any given witness will remember (and report) what actually happened. We'll say that witnesses are reliable if their reliability at least 1/2, and unreliable otherwise. We'll also say that something is probable if its probability is at least 1/2, and improbable otherwise.

Suppose that you know that most witnesses reported what actually happened. Well, then it is improbable that they are unreliable (in the sense I defined above), or equivalently, it is probable that they are reliable.

On the other hand, if you know that most witnesses reported something different from what actually happened, then it is improbable that they are reliable, or equivalently, probable that they are unreliable.

If you don't know what happened, you can't say anything about witnesses' reliability. However, if you assume (or have some other way of determining) the witnesses' reliability, and you know what they have reported, then you can calculate the probability of events having happened. Your analysis fits this case. So you can't say anything about the witnesses' reliability from that analysis.

If you have already have some reason to believe that 0.5 is a good estimate for witnesses' reliability, your analysis would show that (if I understand your post properly) the SBT theory is unlikely. Otherwise your analysis is not enough to demonstrate anything.

On the other hand, if you know that most witnesses reported something different from what actually happened, then it is improbable that they are reliable, or equivalently, probable that they are unreliable.

Yes. I am assuming that if they are unreliable, they are not reliably unreliable. In other words, their recollections will not strongly favour one pattern. If the probability is less than 1/2 then they are reliably wrong (more likely to get it wrong than right) which suggests some common factor is favoring a wrong recollection.

If you don't know what happened, you can't say anything about witnesses' reliability. However, if you assume (or have some other way of determining) the witnesses' reliability, and you know what they have reported, then you can calculate the probability of events having happened. Your analysis fits this case. So you can't say anything about the witnesses' reliability from that analysis.

They could all be reliably unreliable - for example if they observed the colour of something incorrectly because of the colour of a street light illuminating the scene. I am assuming that there was not common factor that would cause them to consistently and independently perceive the events incorrectly.

If you have already have some reason to believe that 0.5 is a good estimate for witnesses' reliability, your analysis would show that (if I understand your post properly) the SBT theory is unlikely. Otherwise your analysis is not enough to demonstrate anything.

Can the analysis not be used to show that there is definitely some statistical significance to 44 out of 50 witnesses recalling the 1..........2....3 shot pattern? - ie. that either they actually heard it or their perceptions were all affected by something that caused them to hear it wrong in the same way?

Yes. I am assuming that if they are unreliable, they are not reliably unreliable. In other words, their recollections will not strongly favour one pattern. If the probability is less than 1/2 then they are reliably wrong (more likely to get it wrong than right) which suggests some common factor is favoring a wrong recollection.

They could all be reliably unreliable - for example if they observed the colour of something incorrectly because of the colour of a street light illuminating the scene. I am assuming that there was not common factor that would cause them to consistently and independently perceive the events incorrectly.

Can the analysis not be used to show that there is definitely some statistical significance to 44 out of 50 witnesses recalling the 1..........2....3 shot pattern? - ie. that either they actually heard it or their perceptions were all affected by something that caused them to hear it wrong in the same way?

AM

Well, my analysis was pretty simplistic. I assumed only two possibilities for what their reports look like: either they report what actually happened, or something else happened. If 44 out of 50 people remember exactly the same thing, and the stories of the others vary, there's definitely a reason to believe the testimony of the 44 over the other options! But neither your nor my analysis are enough to consider a more complicated case.

If there are only two possible oucomes (either they remember 1...2.......3 or 1.......2...3, say), and you assume that there is some parameter, 'reliability', representing the probability that a witness remembers whatever actually happened, then the fact that 44/50 witnesses remember one of the two options is not enough to show that that was what probably happened; for example, if the reliability of each witness is 0.01, and 44/50 remember 1..2......3, then the probability that 1..2......3 actually happened is very small. In particular, it's

[tex]\frac{50!}{44!6!}(0.01)^{44}(0.99)^6.[/tex]

However, I would agree that such low reliability would certainly suggest some external factor explaining its smallness. The point is that in an ideal situation like this, you just can't conclude anything about reliability unless you know what the actual events were.

As I said, this is a very simple analysis, but it is what you did in your post in order to calculate probabilities. There's no direct conclusion about reliability that can be made.

There is so much research available on human inability to recall what we've actually seen, that personally I would doubt any analysis of such data in relation to a case like this (where witnesses are expected to remember something taking place over a few seconds, in a very stressful situation, and where the psychological impact of hearing a gunshot at such an event might well distort perceptions [ie. cause a perceived longer delay after the first shot or something - which might be an explanation for 'reliable unreliableness', for example]! But I don't know anything about that sort of thing, or about this debate in particular, so you should certainly ignore me on this point ).

Well, my analysis was pretty simplistic. I assumed only two possibilities for what their reports look like: either they report what actually happened, or something else happened. If 44 out of 50 people remember exactly the same thing, and the stories of the others vary, there's definitely a reason to believe the testimony of the 44 over the other options! But neither your nor my analysis are enough to consider a more complicated case.

If there are only two possible oucomes (either they remember 1...2.......3 or 1.......2...3, say), and you assume that there is some parameter, 'reliability', representing the probability that a witness remembers whatever actually happened, then the fact that 44/50 witnesses remember one of the two options is not enough to show that that was what probably happened;

It is a little more complicated than that. Many of the 44 who recalled the 1.........2....3 pattern appeared to be very specific and confident. 6 witnesses who remarked on the pattern thought the pattern was 1...2..........3 but none expressed confidence. 9 or 10 thought the shots were about evenly spaced but many of those were distracted and could not be sure. The only outcomes I am concerned with are the 1..........2...3 pattern which 44 witnesses who remarked on the shot pattern recalled and the 1....2.......3 pattern which is required for the SBT.

for example, if the reliability of each witness is 0.01, and 44/50 remember 1..2......3, then the probability that 1..2......3 actually happened is very small. In particular, it's

[tex]\frac{50!}{44!6!}(0.01)^{44}(0.99)^6.[/tex]

However, I would agree that such low reliability would certainly suggest some external factor explaining its smallness.

My point is that if the witnesses had a .99 probability of getting it wrong the same way, there must be some common external factor. You can't get 99% to agree unless there is a reason. It cannot happen by chance.

The point is that in an ideal situation like this, you just can't conclude anything about reliability unless you know what the actual events were.

I think you can - if you can eliminate any common factor that would cause them to get it all wrong in the same way (and feel confident about it). You can say that the result of 44 to 6 for the 1........2....3 pattern is statistically significant in the sense that it did not happen by chance.

There is so much research available on human inability to recall what we've actually seen, that personally I would doubt any analysis of such data in relation to a case like this (where witnesses are expected to remember something taking place over a few seconds, in a very stressful situation, and where the psychological impact of hearing a gunshot at such an event might well distort perceptions [ie. cause a perceived longer delay after the first shot or something - which might be an explanation for 'reliable unreliableness', for example]!

Indeed there are studies. Not many though. They show that witnesses are usually accurate in recalling salient details:Loftus, Eliz. F., Eyewitness Testimony, (Cambridge, MA: 1979), Harvard University Press at p. 25 ff. In the worst case scenario where witnesses are given false suggestions, they are still correct more than 60% of the time. If a witness is confident about a recollection, there is a very high accuracy rate of well over 90% on the salient facts. You can use your own experience here. While I can imagine an external distraction that would cause witnesses to have a poor recollection that they would have no confidence in, I cannot imagine a distraction that would cause them to get it wrong AND make them feel sure they were right.

BTW, I see you are an undergrad in math in Canada. What university are you attending? I went to Queen's; (math and physics '76).

I'm at Carleton. Actually, I am graduating this term (math-physics also!), and next year I'll be at Queen's for graduate school!

Okay, here's how I'd proceed from the current situation: You can use the research cited in your last paragraph to make a reasonable estimate of reliability of witness testimony; say a conservative guess of 0.75 (based on the 90% max and 60% min figures you quoted).

Using that estimate, you can stick to the simple analysis in your post and mine to get a rough upper bound on the probability that something different than 1..2......3 happened (which is what you already did, with a reliability of 0.5; even more conservative, so even better! ). That makes their theory improbable.

My point was only that if you consider reliability to be a free parameter, then you can't get anything out of that sort of simple analysis. You need some outside reference from which to estimate reliability. The reason is that by taking reliability low enough, you can make the probability that the wrong outcome will be reported as close to 1 as you like.

However, a scenario like this could be reasonably analysed, needing only a much weaker outside reference for reliability, by generalizing the method above:

Suppose you have a finite set [itex]S = \{ s_1, ..., s_n\}[/itex] of possible outcomes for some event. You have a collection of M observers, each with reliability [itex]r[/itex].

Now the event takes place, and the outcome is some [itex]s \in S[/itex]. The reliability [itex]r[/itex] means that each observer has probability [itex]r[/itex] of reporting that they saw event [itex]s[/itex], and probability [itex]1-r[/itex] of reporting that they saw something else.

In general this means there's some probability distribution [itex]P:S \rightarrow [0,1][/itex], with [itex]P(s) = r[/itex] and

[tex]1 = \sum_{s^\prime \in S} P(s^\prime).[/tex]

To simplify my analysis I'll assume that P is uniform away from [itex]s[/itex], ie. that [itex]P(s^\prime) = (1-r)/(n-1)[/itex] for each [itex]s^\prime \in S[/itex] not equal to [itex]s[/itex].

Now, the probability that all the witnesses (I am going to approximate your 44 out of 50 by all of them; if I don't do this then everything gets very complicated!) report a particular outcome [itex]s_1 \neq s[/itex] (ie. a particular outcome that is not the real outcome) is just

[tex]P(s_1)^M = \left(\frac{1-r}{n-1}\right)^M.[/tex]

On the other hand, the probability that all the witnesses report outcome [itex]s[/itex] (ie. the real one) is just

[tex]P(s)^M = r^M.[/tex]

Now we arrive at a problem. If

[tex]\frac{1-r}{n-1}>r \ \Longleftrightarrow 1 > nr,[/tex]

then in fact it is more likely that they all report the same wrong outcome than it is that they all report the actual outcome.

Now, here is how you can argue in your case:

The number [itex]n[/itex] of possible outcomes is very large in a situation like this; in other words, there are many, many different possibilities for what people could remember and report. Conservatively, say [itex]n \approx 10[/itex]. Essentially all of the witnesses remember the same outcome (ie. 44 of 50).

In this case, then, in order for it to be more likely that they all report an incorrect outcome than that they all report the correct outcome, we would need

[tex]1 > nr.[/tex]

Using our conservative estimate [itex]n \approx 10[/tex], this would require [tex]r < 0.1[/tex]. However, 10% (the upper bound on r) is an unreasonably small value for witnesses' reliability! Thus, we must conclude it is improbable (in the sense I defined in my first post) that all the witnesses report an outcome that is not the correct outcome (Edit: What I mean here is "it is improbable that, if all the witnesses report seeing the same thing, the thing that they report seeing is not what actually happened.").

Finally, since essentially all the witnesses report the same outcome, this means that it is probable that what they report is what actually happened.

Note that in the case [itex]n=2[/itex] this reduces to your analysis above (only in that case it's easy to analyse things without assuming that all the people see the same thing!). You'll find in that case if [itex]r<0.5[/itex] then it's more likely that everyone reports the wrong outcome than that everyone reports the right outcome, and if [itex]r>0.5[/itex] then it's more likely that everyone reports the right outcome than that everyone reports the wrong outcome. Here, by using the fact that there are actually a great number of variables in peoples' observations, you can argue that everyone reporting the same wrong outcome with confidence would put an unreasonably low upper bound on the witnesses' reliability.

I'm at Carleton. Actually, I am graduating this term (math-physics also!), and next year I'll be at Queen's for graduate school!

Most of my old professors are probably retired now but there may be a few I can warn you about. Good luck.

Using our conservative estimate [itex]n \approx 10[/tex], this would require [tex]r < 0.1[/tex]. However, 10% (the upper bound on r) is an unreasonably small value for witnesses' reliability! Thus, we must conclude it is improbable (in the sense I defined in my first post) that all the witnesses report an outcome that is not the correct outcome.

Finally, since essentially all the witnesses report the same outcome, this means that it is probable that what they report is what actually happened.

I agree with your approach generally. You are saying, and I agree, that the possibility that such a large group of independent witnesses would agree on the shot pattern is extremely unlikely if one's recollection was a matter of pure chance. Therefore, we can say with confidence, that these recollections did not occur by chance and that some factor is operating here to cause that non-random result.

If one had two groups of 50 patients with a disease for which the normal untreated survival rate was 50% but when administering drug A, 44 survived and 6 died, you could conclude that there is something therapuetic about drug A. Someone could argue that actually the drug had the effect of killing 88% but in this case the reverse just happened by chance, but I expect that no one would take it seriously without some pretty good evidence.

Unfortunately, one-time events are not perfectly repeatable. That is why juries are called upon to interpret evidence according to their experience and to apply common sense.

Most of my old professors are probably retired now but there may be a few I can warn you about. Good luck.

Thanks!

Andrew Mason said:

If one had two groups of 50 patients with a disease for which the normal untreated survival rate was 50% but when administering drug A, 44 survived and 6 died, you could conclude that there is something therapuetic about drug A. Someone could argue that actually the drug had the effect of killing 88% but in this case the reverse just happened by chance, but I expect that no one would take it seriously without some pretty good evidence.
AM

That's true. The difference is that in that case you have a solid reference for the normal survival rate. In the situation we were analysing, the difficulty was (or, at least, was initially) that 'reliability' was a free parameter, with no solid outside reference. So you have to be a little bit more careful with the argument!

In analogy: if we don't know the actual survival rate, and it's really 80%, not the assumed 50%, then 44/50 surviving patients wouldn't be particular unusual! In a case like that you wouldn't be able to conclude from the experiment anything about either the medicine's effectiveness or the usual survival rate (which is why I remarked that your initial work wouldn't be quite enough to decide anything ).

Edit: There's a minor problem with the arithmetic in my last post. In fact, because there are [itex]n-1[/itex] non-correct outcomes to choose from, the probability that everyone reports seeing the same non-correct event is

[tex](n-1)\left(\frac{1-r}{n-1}\right)^M.[/tex]

That changes the [itex]1>nr[/itex] condition to

[tex]1 > \left((n-1)^{1 - 1/M} + 1\right)r,[/tex]

which goes to the old condition in the limit of large M (ie. a large number of witnesses). The quantitative difference for the argument is minimal; For example with n=10,

[tex]9^{1-1/50} + 1 \approx 9.6,[/tex]

so things will work out essentially the same way for large enough M relative to n (ie. a large number of witnesses relative to possible outcomes - practically, even when [itex]n \approx M[/itex] things work out pretty well the same way - and note, when [itex]n=2[/itex] it still reduces to the same thing as before, [itex]1<2r[/itex]).

That's true. The difference is that in that case you have a solid reference for the normal survival rate. In the situation we were analysing, the difficulty was (or, at least, was initially) that 'reliability' was a free parameter, with no solid outside reference. So you have to be a little bit more careful with the argument!

In analogy: if we don't know the actual survival rate, and it's really 80%, not the assumed 50%, then 44/50 surviving patients wouldn't be particular unusual! In a case like that you wouldn't be able to conclude from the experiment anything about either the medicine's effectiveness or the usual survival rate (which is why I remarked that your initial work wouldn't be quite enough to decide anything ).

My only comment would be that juries can rely on their own experience to determine whether it is reasonable to assume that witnesses would not have a significantly greater chance of getting an observation wrong (and feeling confident they were right), than in recalling it correctly. If they start with the assumption that witnesses have a greater than even chance of getting right, then as a matter of statistics there is only one conclusion to draw from the fact that 44 recalled the 1.........2...3 pattern: ie that they observed it correctly.

BTW, I believe that the Single Bullet Theory is wrong but that Oswald did all the shooting. The idea that the SBT is needed to dispell the need for two shooters is wrong, if the shot pattern witnesses were correct. If they were right, there was only one shot before the car appears from behind the sign and JFK is seen to be reacting to being shot in the neck. It simply means that the governor was not hit in the back at that point. Although he appears to be reacting to something, he said he reacted to the first shot because he recognized it as a rifle shot and turned around to see JFK. He turns around and looks back toward JFK up to the point at which the witnesses put the second shot (frame 270 or so).

I believe that the Single Bullet Theory is wrong but that Oswald did all the shooting.

Hi Andrew. I respect the probabilities as discussed, but before I address them I must challenge your last observation. The two statements are not compatible. In other words, to say that Oswald did all the shooting does require the single bullet theory.

Here's why. Whether Oswald did all the shooting or not, the following propositions are generally undisputed:

Those who maintain that Oswald did all the shooting usually accept each of these statements as fact.

The single bullet theory has always been used to account for b) and c). If the theory is wrong, then either one of the five propositions has to go or Oswald did not do all the shooting.

This was precisely the problem encountered by the Warren Commission. If Connally and Kennedy were hit by separate bullets, there had to be a second assasin. That was how the single bullet theory was born.

You have statistically shown how unlikely it is that eye witnesses could collectively misperceive the same event. If valid, you have demonstrated that the single bullet scenario is unlikely to have happened. By the same argument, you have demonstrated that in all probability Oswald did not do all the shooting.

If you wish to state the opposite - that Oswald probably did all the shooting - you have three choices:

1) Find evidence that allows Oswald to do all the shooting without the single bullet theory.

2) Accept that the single bullet theory is indespensable and overturn your eye witness analysis, thereby establishing a higher probabilty for the single bullet theory.

3) Accept that the single bullet theory is indespensable and find evidence that independently increases the probabilty of the single bullet theory or indeed that Oswald did all the shooting. This independent probabilty must be sufficiently high so that even when combined with the anomolous eye witness statistics, the result will be above 1/2 for the single bullet theory.

I concede that 2) and 3) may remain open questions, but defenders of the lone gunman have always had to accept that 1) is impossible.

Hi Andrew. I respect the probabilities as discussed, but before I address them I must challenge your last observation. The two statements are not compatible. In other words, to say that Oswald did all the shooting does require the single bullet theory.

Here's why. Whether Oswald did all the shooting or not, the following propositions are generally undisputed:

I would agree with a, b, c and d. But not only is there no evidence that a shot missed the limo, there is evidence that all three hit. The Warren Commission has a whole section entitled "the shot that missed" but could not explain which shot it was. There is strong evidence against the first, second or third shot missing, as they point out.

Until fairly recently most SBT supporters preferred the first shot SBT in which the shot hit JFK and Connally but Connally did not feel being hit in the back (contrary to what he said - he said he heard the first shot and turned right to see JFK, couldn't see him and then felt a huge punch-like impact to his back below his shoulder blade). Now the SBT supporters prefer the second shot SBT and say the first shot missed.

Those who maintain that Oswald did all the shooting usually accept each of these statements as fact.

Not everyone. Not me, in particular.

The single bullet theory has always been used to account for b) and c). If the theory is wrong, then either one of the five propositions has to go or Oswald did not do all the shooting.

Yes. e) has to go. If e) goes, there is no problem provided Connally is not hit in the back until much later - well after frame 230 where most people think he was hit. But there is abundant evidence that he was not hit at that point.

This was precisely the problem encountered by the Warren Commission. If Connally and Kennedy were hit by separate bullets, there had to be a second assasin. That was how the single bullet theory was born.

Only if JFK and JBC were hit by frame 230. My point is that the 1.......2...3 shot pattern tells you that there was only one shot to that point. The first shot SBT is the only possibility. And the Connallys (and others) nix that possibility.

You have statistically shown how unlikely it is that eye witnesses could collectively misperceive the same event. If valid, you have demonstrated that the single bullet scenario is unlikely to have happened. By the same argument, you have demonstrated that in all probability Oswald did not do all the shooting.

No. The shot pattern tells you that there was only one shot before the midpoint between the first and last. That midpoint is at least as late as z240. If the first shot was around z200, the midpoint is z256 (the last being at z313). So long as the last shot was before z272 or so, there is not a problem for Oswald to have made three shots. I put the second shot at z271 striking JBC in the back.

If you wish to state the opposite - that Oswald probably did all the shooting - you have three choices:

1) Find evidence that allows Oswald to do all the shooting without the single bullet theory.

2) Accept that the single bullet theory is indespensable and overturn your eye witness analysis, thereby establishing a higher probabilty for the single bullet theory.

3) Accept that the single bullet theory is indespensable and find evidence that independently increases the probabilty of the single bullet theory or indeed that Oswald did all the shooting. This independent probabilty must be sufficiently high so that even when combined with the anomolous eye witness statistics, the result will be above 1/2 for the single bullet theory.

I concede that 2) and 3) may remain open questions, but defenders of the lone gunman have always had to accept that 1) is impossible.

And the witness evidence tells you that your premise (that 1 is impossible) is false.

The key is realizing that JBC is reacting (as he said) to the sound of the first shot from z228-271) and not to being hit in the back by it. The witness evidence tells you that there was only one shot to that point. And the Connallys (and a lot of other evidence) make it pretty clear that it did not hit the Governor in the back.

Ok, for our purposes, I will not treat the missed shot as an a priori fact. Whether there is evidence for it is a separate subject, but we can certainly throw it out as a given proposition. (However, a missed shot is automatically required if the SBT is true and Oswald did all the shooting - which is precisely what we're debating.)

Instead, I will confine myself to what originally faced the Warren Commissioners and many researchers since. This is the timing problem posed by the Zapruder film.

Here are facts that we can agree on:

1) Each Zapbruder corresponds to 1/18th of a second.

2) The minimum loading time for the Manlicher Carcano rifle was 2.3 seconds - or 42 frames.

4) Between frames 211-233, Kennedy is clearly seen physically reacting to having been shot. Connally shows no sign of any such reaction. By his own testimony, he felt the impact when a bullet hit him.

5) The SBT is definitely not required for Oswald to have done all the shooting so long as Connally was hit more than 42 frames after Kennedy. In this scenario, two bullets could have come from the same rifle.

6) The SBT is defintely required for Oswald to have done all the shooting if Connally was hit less than 42 frames after Kennedy. In this scenario, two bullets could not have come from the same rifle.

We both agree that Connally shows no sign of being hit between frames 200 and 233. This is a familiar argument against the SBT. The crucial period is the eight frames between 234 and 242. Is it a myth that Connally was reacting to a shot at this time? If so, you have made your case - Connally was not hit less than 2.3 seconds after Kennedy. If Connally was hit as late as frame 271, Oswald had enough time - approximately 4 seconds to reload.

Based on the Zapruder film, however, the wide consensus is that Connally was reacting to a shot between frames 233 and 242. For the Warren Commision, it was precisely this timing problem that required the unwanted SBT to rule out a second gunman.

However, I accept the implausiblity of the single bullet theory. I accept that its improbality is born out not only by your own eye witnesss analysis, but by a wealth of other data. I accept that, quite apart from Connally's own testimony, the severity of his five wounds also makes a delayed reaction betwen frames 200 and 233 unlikely.

Is it reasonable to assume that a witness's report or reliability is independent of the others?

It depends on the circumstances. Witnesses are susceptible to suggestion but normally it is not a huge factor (see: Loftus, Eliz. F., Eyewitness Testimony, (Cambridge, MA: 1979), Harvard University Press at p. 25 ff). There was a lot of publicity of the number of shots (3) and people who were unsure of the number of shots but thought there were 3 might be more confident about saying 3 shots if they had been exposed to media reports before giving their statements. But there was almost no publicity about the shot pattern. There is nothing to indicate that the FBI or Dallas Sheriff's officers were suggesting any preferred shot pattern.

2) The minimum loading time for the Manlicher Carcano rifle was 2.3 seconds - or 42 frames.

I am not sure it is that exact. It is close. The FBI, without a lot of practice with the rifle, fired 3 shots in 4.45 seconds with the iron sights and 3 shots in 4.6 seconds using the telescopic sight. So it is likely between 2.2 and 2.3 seconds or 40-42 frames.

It could be a few frames earlier but Oswald would be shooting while JFK was just emerging from under the oak tree. The oak tree did not completely obscure him, however. I put it at z198-202.

It is possible to bracket first shot using witnesses. For example, all the occupants of the VP car said they had turned the corner and were going down Elm St. when the first shot was heard. It is still in the turn at z180. Occupants of the VP followup car said they were completing the turn and were along side the TSBD. They are still pointing toward the TSBD at z191. Hugh Betzner said he took his z186 photo and started winding his camera to take another when the first shot was heard. Phil Willis said he took his z202 frame at the instant he heard the first shot. TE Moore said JFK was opposite the Thornton Freeway sign when the first shot occured (that corresponds to frame z200). Mary Woodward described yelling to the President and he and Jackie turned to smile and wave at them. He turned forward and then there was a "horrible ear-shattering noise". We can see his turn, smiles, waves and turn forward exactly as she described on Nov 22/63. He turned forward between z197-198.

4) Between frames 211-233, Kennedy is clearly seen physically reacting to having been shot. Connally shows no sign of any such reaction. By his own testimony, he felt the impact when a bullet hit him.

Agreed. He can't be seen from z211-224, as he is behind the Stemmons freeway sign.

5) The SBT is definitely not required for Oswald to have done all the shooting so long as Connally was hit more than 42 frames after Kennedy. In this scenario, two bullets could have come from the same rifle.

Agreed.

6) The SBT is defintely required for Oswald to have done all the shooting if Connally was hit less than 42 frames after Kennedy. In this scenario, two bullets could not have come from the same rifle.[/B]

42 is pretty close to the limit. Oswald practised a lot using his bolt action, according to Marina. As I say, I think it could have been done in 40 frames or 2.2 seconds.

We both agree that Connally shows no sign of being hit between frames 200 and 233. This is a familiar argument against the SBT. The crucial period is the eight frames between 234 and 242. Is it a myth that Connally was reacting to a shot at this time?

It is not exactly a myth. The Connallys themselves thought he was hit about z230 (their opinion was based on seeing the film and then reviewing prints of the frames). But it does not fit with the evidence. Connally said he was hit after turning back to see JFK. He does not begin to turn back before z235. He turned back from z235-270. He said that JFK was not in his normal place so he decided to turn back to his left when he was hit in the back. His wife said he was turned right when hit. She also said he said "oh, no, no, no" BEFORE he was hit. He is saying that from z242-248 or so. (JBC said he recognized the sound as a rifle shot and immediately feared an assassination was unfolding).

Nellie also said that she never looked back into the back seat after the second shot. She is staring at JFK up to z255. Altgens said his z255 photo was taken after the first and before any other shots. Greer said he turned back immediately upon hearing the second shot. He turns at z280.

If so, you have made your case - Connally was not hit less than 2.3 seconds after Kennedy. If Connally was hit as late as frame 271, Oswald had enough time - approximately 4 seconds to reload.

That is my case, exactly. JBC was hit by the second shot. The witnesses overwhelmingly put the second shot after the midpoint between 1 and 3 and other evidence puts it at z271. That is 42 frames before z313. Oswald simply fired the last two shots as quickly as he could as the car was moving away, which is what all the witnesses heard.

Based on the Zapruder film, however, the wide consensus is that Connally was reacting to a shot between frames 233 and 242. For the Warren Commision, it was precisely this timing problem that required the unwanted SBT to rule out a second gunman.

Actually, the FBI originally thought JBC was hit around z275. But an FBI expert (Robert Frazier) said that he could not have been hit after z240 because JBC was turned too far right for a bullet to have gone through his chest the way it did without changing direction. (but he did not express an opinion whether the bullet changed direction).

However, I accept the implausiblity of the single bullet theory. I accept that its improbality is born out not only by your own eye witnesss analysis, but by a wealth of other data. I accept that, quite apart from Connally's own testimony, the severity of his five wounds also makes a delayed reaction betwen frames 200 and 233 unlikely.

"Reliably unreliable" performance for individuals vs. groups is (or, used to be) a major factor in assessing risk; the more people involved the higher the probability that you'd get an "unfavorable" outcome ("too many cooks" effect). Do you throw out a witness for a single error in description of the sequence of events? Do you examine each item in the sequence independently of the others? What happens to "sense of time" following "startle" reaction? Is there "information" in the eyewitness accounts? Yeah, probably. Is it worth the effort necessary to dig it out? Not likely.

As already stated, if both Kennedy and Connally were hit within less than 42 frames and if Oswald did all shooting, then the SBT is fact.

Now, in your analysis, you maintain the following:

a) the first shot wounded Kennedy at around frame 200
b) the second shot wounded Connally at around frame 271
c) the third shot killed Kennedy at frame 313

Ok, so 4 seconds have elapsed after the first shot wounded Kennedy. This rules out any need for a SBT to account for shots 1 and 2. However, you now have to explain the very narrow gap between shots 2 and 3. As you pointed out, frame 271 is exactly 42 frames before Kennedy's fatal shot at frame 313.

You have given Oswald close to the minimum reload window of 2.22 seconds in which to fire the fatal shot on target at a moving car. Although this is not physically impossible, it is highly improbable. The minimum reload does not include the time required to aim the rifle. For this, I would say you must add a minimum of half a second - about 8 or 9 frames. Without this, we are talking about a blind shot which by coincidence hit the bullseye.

I would have to conclude the following: if, as you maintain, Connally's shot and Kennedy's fatal shot occured within less than 50 frames, it is highly improbable that Oswald's rifle fired both of them. This improbability can only be dispelled by a second gunman.

Whereas if we put the minimum time window between shots 1 and 2, we have a more probable version of your lone-gunman-without-the-STB:

a) the first shot wounded Kennedy at frame 200
b) the second shot wounded Connally at frame 242
c) the third shot killed Kennedy at frame 313

Here, after wounding Kennedy, Oswald reloads instantly and fires blindly. He hits Connally by chance. He then fires again with pin-point accuracy, this time taking about four seconds to reload - a much more plausible scenario. I beleive this closely ressembles the FBI's first reconstruction.

Unfortunately it contradicts both your analysis of the witnesses and the many researchers who percieve that the film shows Connally reacting before frame 242.

To dissovle these contradictions and remove all the improbabilities we've talked about, there is one simple answer which is the most plausible in my opinion - the more-than-one-assassin scenario.

A two gunman scenario would allow all those researchers to be correct that Kennedy and Connally were wounded within frames 200 and 242, without resorting to the improbable SBT. A two gunman scenario would also allow all your cited witnesses to be correct that Connally was hit at frame 271, without making the fatal shot at frame 313 a lucky hit.

The second most plausible scenario is one gunman and the SBT. This has problems that we've already covered- not least your witness analysis. But it has an internal consistency.

I'm forced to conclude that the least plausible scenario is one gunman but no SBT. It means researchers are all wrong that Connally was shot before frame 242 - a co-incidence that needs explaining. Moreover, if your witnesses are correct that he was shot at frame 271, then Oswald's third shot was not aimed but nevertheless hit the target - another co-incidence that needs expaining.

As already stated, if both Kennedy and Connally were hit within less than 42 frames and if Oswald did all shooting, then the SBT is fact.

Now, in your analysis, you maintain the following:

a) the first shot wounded Kennedy at around frame 200
b) the second shot wounded Connally at around frame 271
c) the third shot killed Kennedy at frame 313

Ok, so 4 seconds have elapsed after the first shot wounded Kennedy. This rules out any need for a SBT to account for shots 1 and 2. However, you now have to explain the very narrow gap between shots 2 and 3. As you pointed out, frame 271 is exactly 42 frames before Kennedy's fatal shot at frame 313.

You have given Oswald close to the minimum reload window of 2.22 seconds in which to fire the fatal shot on target at a moving car. Although this is not physically impossible, it is highly improbable. The minimum reload does not include the time required to aim the rifle. For this, I would say you must add a minimum of half a second - about 8 or 9 frames. Without this, we are talking about a blind shot which by coincidence hit the bullseye.

No. The 2.3 seconds is the time required to shoot, hit the target at 225 feet, reload, reaim, shoot AND hit the target at 275 feet. The FBI recreated the shooting using Oswald's rifle. They placed targets at between 175 and 275 feet. They had different shooters shooting sets of 3 shots. All shots hit within 9 inches of the target and times were as low as 4.45 seconds (average 2.25 seconds between shots). You should read http://www.dufourlaw.com/jfk/shot_pattern_evidence.pdf" that I cited:

The Warren Commission found that a minimum of about 2.3 seconds was required to fire, reload aim and fire again using Oswald’s rifle.(http://www.history-matters.com/archive/jfk/wc/wr/html/WCReport_0061a.htm" [Broken])."​

Not blindly. Agent Hickey, who was riding in the car behing JFK said he saw JFK's hair fly up on the second shot which indicated to him that it just missed. Of course, if that is correct, then Gov. Connally was hit by it because he was in front of him. You can see JFK's hair fly up beginning at z271.

He hits Connally by chance. He then fires again with pin-point accuracy, this time taking about four seconds to reload - a much more plausible scenario. I beleive this closely ressembles the FBI's first reconstruction.

The only problem is that your scenario doesn't fit the facts. The witnesses were very consistent that the last two shots were very close together. The FBI's first report stated that the second shot struck JBC at about frame 275.

To dissovle these contradictions and remove all the improbabilities we've talked about, there is one simple answer which is the most plausible in my opinion - the more-than-one-assassin scenario.

No. There is strong evidence that all shots came from the same location. There were witnesses who saw the rifle and heard 3 shots. There is strong evidence that three shots were fired from that gun at that location. Three men 10 feet below on the 5th floor attested to that. They even heard the shells hit the floor (the floor was partly open and dust from the 6th floor fell on one of their heads). Three shells were found on the floor. There is absolutely no evidence of any other shooter or gun or evidence that is inconsistent with Oswald firing all three shots.

I'm forced to conclude that the least plausible scenario is one gunman but no SBT.

Is this based on the view that Oswald needed 50 frames between shots? That is not the evidence.

Any scenario is improbable. There are an infinite number of ways an event can occur. What makes the improbable probable or certain is evidence.

Here we have strong evidence of three shots, three hits. We have no evidence of only two hits. We have only a theory which requires rejecting vast amounts of evidence. I contend that not only is that evidence credible, it is impossible that it is wrong.

"Reliably unreliable" performance for individuals vs. groups is (or, used to be) a major factor in assessing risk; the more people involved the higher the probability that you'd get an "unfavorable" outcome ("too many cooks" effect). Do you throw out a witness for a single error in description of the sequence of events? Do you examine each item in the sequence independently of the others? What happens to "sense of time" following "startle" reaction? Is there "information" in the eyewitness accounts? Yeah, probably. Is it worth the effort necessary to dig it out? Not likely.

I don't think reliability of people to perform a task correctly without error has much to do with this. The reliability of a person to perform a task is not 100% so we know they are bound to fail given enough time. But the issue here is not whether a single witness might be wrong - or fail. The issue is whether 44 witnesses will all fail at the same time in the same way. It is analogous to having 50 workers who perform properly 90% of the time and 44 of them all failing in the same way at the same time.

There is no question that a witness can be unreliable. On the other hand, witnesses are certainly capable of seeing something and recalling correctly the salient details of what they observed. Elizabeth Loftus in her book Eyewitness Testimony (http://www.dufourlaw.com/jfk/loftus.PDF") refers to a study which shows that witnesses will recall the most salient details with an accuracy and completeness of 98% (p. 27). It is important to know how to tell the difference between an unreliable recollection and a reliable one.

My point is that when a witness' recollection of a simple fact is independently corroborated by 43 other independent and unbiased witnesses who all had similar recollections, there is a common reason. The only common reason in this case that makes any sense is that they all observed the same thing.

Is there "information" in the eyewitness accounts? Yeah, probably. Is it worth the effort necessary to dig it out? Not likely.

This is the nub of the issue. I say that if 44 out of 60 witnesses agree on a simple fact relating to an event, for which there are at least three possible choices (eg. last two bunched, first two bunched, none bunched), something is driving that agreement. That is the information you can draw from this.

The only question then, is what is driving the agreement? If you can eliminate collusion or bias, there is only one thing that could be driving it: the event actually occurred that way. I think that is worth digging out.